`Math 131 Week in Review for sections 4.2, 4.3, and 4.6
J. Lewis
1. Find all local max, local min and inflection points of the function with the given derivative.
3
a)
f ' ( x )  ( x  1 )( x  2 )
b)
f ' ( x )  ( x  1) ( x  2 )
c)
f '(x) 
2
3
 6x
2
(3  x )
2
2. Find the absolute max and absolute min of f(x) on the given interval.
0  x
a)
 x ln x
f (x)  
 0
b)
f ( x )  ( x  1) ( x  2 )
x  0
2
and on
on
[0, e]
on
[  2 ,2 ]
[  3,2 ]
3. The table shows some values of f ‘ and of f “ for a function.
What does the 2nd derivative test say about the graph of the function in each case?
x
x1
x2
x3
x4
f '(x)
0
0
1
0
f "(x)
0
3
2
1
4. From text by Barnett, Ziegler and Byleen. A university student center sells 1,600 cups of
coffee per day at a price of $2.40.
A market survey shows that for every $0.05 reduction in price, 50 more cups of coffee will be
sold. How much should the student center charge for a cup of coffee in order to maximize
revenue? Let q be the quantity of cups sold per day and p the price per cup. Then revenue is qp.
Find the line p(q) = mq +b and then R(q)=q(mq+b). Maximize R(q).
Problems from pages 392 - 394 in the text by Warner/Costenoble
5. #38 A company manufactures cylindrical metal drums with open tops with a volume of
1 cubic meter. What should be the dimensions of the drum in order to use the least amount of
metal in their production?
6. #42 Vanilla Box Company is going to make open topped boxes out of 12x12 inch rectangles
of cardboard by cutting squares out of the corners and folding up the sides. What is the largest
volume box it can make this way?
7. #46 American Airlines requires that the total outside dimensions (length + width + height) of a
carry-on bag not exceed 45 inches. Suppose you want to carry on a bag whose length is twice its
height. What is the largest volume bag of this shape that you can carry on an American flight?
8. #50 UPS will accept only packages with a length of no more than 108 inches and length plus
girth of no more than 165 inches. Assuming that the front face of the package is square, what is
the largest volume package that UPS will accept?
9. A person wants to enclose a rectangular area of 600 sq feet with a fence. The front of the fence
will be made of a material costing $20 per linear foot. (Assume the height is all the same so the
fence is priced by the foot not the square foot.) The other three sides will be made from a
material costing $10 per foot. Find the dimensions of the rectangle that will minimize the cost of
the fence.
10 and 11 are from the text Applied Calculus by Soo T. Tan 9th ed
10. When organic waste is dumped into a pond, the oxidation process that takes place reduces the
pond’s oxygen content. However, given time, nature will restore the oxygen content to its natural
level. Suppose the oxygen content t days after organic waste has been dumped into the pond is
given by f ( t )  100 (1 
4t
) % of its normal level.
2
t  4
a) When is the level of oxygen content lowest?
b) When is the rate of oxygen regeneration greatest?
11. The amount of nitrogen dioxide present in the atmosphere on a certain May day in the city of
Long Beach is approximated by A ( t ) 
136
 28
0  t  11
2
1  . 25 ( t  4 . 5 )
With t=0 corresponding to 7 a.m. Determine the time of day when the pollution is its highest
level.